The present invention relates to a method for gradually deforming, partly or totally, realizations or representations of a heterogeneous medium such as an underground zone, which all are the expression of a Gaussian or similar type stochastic model, i.e. including an underlying Gaussian model.
The method according to the invention finds applications notably in the construction of a stochastic model of an underground formation, constrained by non-linear data.
Examples of use of Gaussian models for modelling the subsoil structure are described by:
Journel, A. G. and Huijbregts Ch. J.: xe2x80x9cMining Geostaticsxe2x80x9d, Academic Press, 1978, or
Matheron, G. et al.: xe2x80x9cConditional Simulation of the Geometry of Fluviodeltaic Reservoirsxe2x80x9d, paper SPE 16753; SPE Annual Technical Conference and Exhibition, Las Vegas, 1987.
Constraint of stochastic models by a series of non-linear data can be considered to be an optimization problem with definition of an objective function assessing the accordance between data measured in the medium to be modelled and corresponding responses of the stochastic model, and minimization of this function. A good deformation method must respect the following three characteristics: a) the deformed model must be a realization of the stochastic model; b) it must produce a regular objective function that can be processed by an efficient optimization algorithm; and c) the deformation must be such that the solution space can be entirely covered.
The method according to which values are exchanged at two points of a realization of the stochastic model, i.e. a realization or representation of the medium studied that is the expression of the model, can be cited as an example of a known deformation method. This method is used in the technique referred to as xe2x80x9csimulated annealingxe2x80x9d known to specialists. An example thereof is described by:
Deutch, C.: xe2x80x9cConditioning Reservoir Models to Well Test Informationxe2x80x9d, in Soares, A. (ed.), Geostatistics Troia""92, pp. 505-518, Kluwer Academic Pub., 1993.
As an arbitrary exchange breaks the spatial continuity of a stochastic model, it is necessary to include a variogram in the objective function, which makes optimization very tedious.
Another known method is the method referred to as pilot point method, which is described for example by:
de Marsily, G. et al.: xe2x80x9cInterpretation of Interference Tests in a Well Field using Geostatistical Techniques to Fit the Permeability Distribution in a Reservoir Modelxe2x80x9d in Verly, G. et al. (ed.), Geostatistics for Natural Resources Characterization, Part 2, 831-849, D. Reidel Pub. Co, 1984.
This method essentially consists in selecting a certain number of points of an initial realization (representation), in calculating the derivatives (sensitivity coefficients) of the objective function in relation to the values at these points, and then in modifying the values at pilot points in order to take account of these sensitivity coefficients. A new realization is formed by the known method referred to as conditional kriging. It may follow therefrom that a change in the values at the pilot points, by using the sensitivity coefficients, leads to an undue change in the variogram even if conditional kriging is used afterwards, in particular when the number of pilot points becomes great.
The method according to the invention allows to carry out at least partial gradual deformation of the realization (or representation) of a Gaussian or similar type stochastic model of a heterogeneous medium such as an underground zone, constrained by a series of parameters relative to the structure of the medium. It is characterized in that it comprises drawing a number p at least equal to two of independent realizations of at least part of the selected medium model from all the possible realizations, and linear combination of these p realizations with p coefficients such that the sum of their squares is equal to 1. This linear combination constitutes a new realization of the stochastic model and it gradually deforms when the p coefficients are gradually modified.
More generally, the method can comprise several iterative stages of gradual deformation, with combination at each stage of a composite realization obtained at the previous stage with q new independent realizations drawn from all the realizations.
The method allows gradual deformation of a model representative of the medium while modifying the statistical parameters relative to the medium structure.
The method also allows to perform individual gradual deformations of various parts of the model while preserving continuity between these parts (i.e., for example, reproduction of a variogram linked with this model).
The coefficients of the linear combination preferably are trigonometric functions.
A Gaussian model or a model similar to the Gaussian type can be used as the model: lognormal model, truncated Gaussian model, etc.
In relation to the prior art, the method according to the invention is founded on a more solid theoretical basis for stochastic model deformation. It also makes it possible both to modify the parameters (often a priori uncertain values) of variograms and to deform a realization of a stochastic model. Finally, thanks to the possibility of individual deformation of various parts of the model, the method according to the invention gives greater flexibility and efficiency to the operator for adjustment of the stochastic model to the field reality. The method according to the invention thus allows to establish a connection between stochastic model adjustment and deterministic model adjustment by zoning conventionally used by reservoir engineers.